Even working directly for a large company is a gamble for someone with a visa. It's one of the little secrets of H1Bs, Green Cards and consulting: Go though a consulting firm, and you get hit by less incidental layoffs (since you are just capex), and even if you are not renewed, the consulting firm will just keep you on the roster (although probably unpaid), while they hand you another gig. Compared to the ticking bomb caused by a layoff, it makes those places attractive from a safety perspective.
This is not really a valley thing either: Here in the midwest, we have companies that are 50+% indian consultants, and companies that are 95% white, with very little in between, and it's all about whether H1Bs work there, directly or consulting.
My former company (AeroFS) was one of the few I'm aware of that was willing to hire good talent even if it required a visa.
Also, I think that asking someone from India (taking that country as an example) to move to the US for less than market wage to toss the dice on a startup is... a challenge. That startup is less marketable on their resume than BigCorp if they want to return to India and get hired by another company or join an outsourcing company.
